sampexam2bsoln

sampexam2bsoln - STAT 8260 Exam 2 — Tuesday April 10 SHOW...

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Unformatted text preview: STAT 8260 Exam 2 — Tuesday, April 10 SHOW ALL WORK Name: (4) 24‘ e. 1. Suppose we take a random sample of n r» 6 adult father-son pairs and we measure the height of each man. Suppose that 3 of the sons in this sample were first-born children in their family and the other 3 happened to be second-born. Let (yij, as”) by the (son’s height, father height) pair of measurements corresponding to the jth son of the ith birth order, 2' = 1, 2, j = 1,2, 3. Consider the following five models for such data. yij 0H + eij ) ) yij =a+fi$ij+eij yij = 0.1 + £3,113.55,- + 81;]; ) yij:#+051r(i:2)+fi$ij+eij ) yij=H+a1(i=2)+fi$,jI(i:l)+7x,jI(i=2)+eij where I denotes an indicator variable taking the value 1 when the condition inside the parentheses is true and the value 0 otherwise. All models have the same assumptions on the eijS: 811,...623 trad N(0, 02). a. (8 pts.) Which, if any, of these models are equivalent. ((1?) (V) W1 éfVIl’fi/‘fii )g 1%)0 are a” yw gay/J M54 aim % m e/ marézanfl 4// 5- m/d, M) (Luz WAIOL one! Awe >g/am-c CD/zmm Vacs 24/ f MAJ! 7%: 7; J' MV‘Z‘ gar/u %’" MoJel Veil-{(2.3 2 [WI , 0?, a.) 04, 74; >46 ijmyf,rt5/oeq4m9. flock, (if) Eye-aids a 5245/! fej/gS’fiBn /4¢ /Lutm~o4 lavécuf/i/EJIA/fi/i) fi‘ q//fvéj‘a713, /7-ao/el (at) a/Aur j» reg/05% Ad, 06-53% 74 5m? /J “Add/9% 75/ 5M)” 2 - #7de (5") 5 ear/u 2 alt/Kai «aka/797k ‘49 /M+o( A,“ flaw/U f9“) iuf‘c‘reagnb/ “3/6 le‘tdn. fé/L /“ J./ (v) Aka not/fl (:22) 57°04; c/erjkc/Ihfiyé/k b1) fIAJ/Jl fit J / V4- % Whig 7/, ,15 2 gnu/33 4.4) Jar/{457’ Singflfl “'3 Y) ) 1 2. éfw S- b. (9 pts.) Write down model (iii) in vector/ matrix notatiOn. ‘3“ I o Xu 0 “(I 2“ ll.- “3‘1 o X” 0 dz 1L CI? “3:? : i o X13 0 f: 62‘ 22»! O l 0 X“ 824. L o l 0 K11. fl; ‘3‘" 0 t (9 XL} 613 3’55 c. (9 pts.) Under the maintained hypothesis that model (v) holds, express the hypothesis that H0 : {model (ii) holds}, in the form of the general linear hypothesis on the parameters of model fl 0 l 0 O - C :1 Il, pale/€— C 2 VB “ °’ 2f d. (9 pts.) Based upon the results in the table below, compute an appropriate test statistic (give me its numeric value for these data) and give its reference distribution for testing the hypothesis in part Model MSE R2 (ii) 9.9268 .0432 (v) 3.2262 .8445 [federal W? 74 Ann/v7: F) a 6; ‘ A?” : (LIE-7W} (\- T:[A, n-k") ; (.Ql‘lfiflBQ/L F/ZIZ) UAJQ/ <1, .K‘Hfl/g—z— :) : 5.7;, 2. Consider the regression model 911 1 611 9'12 1 612 2 + @121 fi 1 621 922 1 622 = [3.14 + e where (811) N N (0,0? (I p)) independent of (621) N N(0,0'§I) where 612 p 1 622 03 = 20? and p = .5. a. (13 pts.) Derive a simple, non-matrix formula for BBLUE, the best linear unbiased estimator of fl in this model. I , g- 0 o 6~N{o,6,"’l/) aim V: 29 cf /\ a 40 0 ‘9 2 r Engf—(S % agn‘flr ()(TV “14,1 4 ‘f ' " O -r 3- s .1 .1) Ami/ab,“ v= a w .1. 0 “‘4— 3 9 fl; ‘4) (m> a: fi —l l i (1'. 3 .3 j (J‘TV-JJIJ’VJ " % E 1‘: j 1 '72 ¥)/l1)/'7 _ _'_3_ :5 4' N ‘35" b. (13 pts.) Assuming the model is correct, compute the ratio Y \ =fl=‘{ val:(g‘.) WWWBLUE)a where 3].. = fizz-2:1 23:1 yij is the sample mean. ‘57.. w 245 am any/M /) MK WM“ _. - o "\ -—- " ' ' Wm): We"ersz =91) JEWEL). 2, . ... _L. I -... 3 .3. \ ) J VA” 7:) LJZJZ)‘J— :31. ._ 1' 7' M, _(5_\ /z _ Sr 03 Tle‘l 30—?3/¥ A __ TV : \(oJ(/3I3Lwr) ’ 0—: (X X I J a —— J 3 W 3 Varbfi 3 I H: __ _,___._ :1 4 2 a 3— "" Q may» a; g, 7:1 4075'“ > I 3. Consider the classical linear model y = Xfi + e, e N Nn(0,a21n) where X is n x (Is + 1) with rank(X) = k + 1 < n. Suppose this model is fit to data y yielding the ordinary least squares regression parameter estimate [3 and. MSE, 32. We wish to form a prediction interval for an additional observation yo conforming to this model. That is, yo = xiii? + 60 Where am is the a vector of explanatory variables corresponding to yo (x0 can be considered as an additional row of X) and 60 w N(0, 02) independent of e. (13 pts.) For 330 x x313, show that 310-330 8V 1 +XE(XTX)_1X0 N150” W k — 1). j \ 1. 4 1. l -. film/UK;— if” 5W gI-vN/fl, Mm Ma‘s ’ ’ 4. Consider the classical linear model yzxfi‘l'e: ewNn(070-2In) Where X is n x p with rank(X) = p < n. Define (33 = SSE/(n—E), an estimator of 02. Recall the mean square error of an estimator T of a parameter 9 is E{(T ~6)2} . . 2 but can also be represented as varlance + blas . a. (13 pts.) Find the mean square error of 6'3, for an arbitrary value of the constantfl. ,_.'7 Var 332;): (All); WM?“ ’Cm‘é’l ‘- ~—-’ Erma: ewe/(KW); HM _.- @rl)2 =0 1 Z 1:5 (:5 — (rt-if “we” 9“” 2 ‘— fi (1- 2. ’_ ,BFM(E¢2): ,Bms(n—o §§c):[(m_£$3 (5' re n") (IA‘ 2 z A“ =S“ :_ 11 £152 (5 : n“! M T - NZ __ n— L! (1—. Z EgzZOj-I-O” FEED b. (13 pts.) Find the value of E that yields the estimator Er? with the smallest _ ‘i mean square error f ~2.) ’ W + q r 20-90143) (“015 4' 4 52+ H0“ (11’?) 9470179} _.| : O (“gigs M? W} (WMMWJL 5’ 2m» 3 l +(n-f>(nnufe" :0 1:) :2 + (m? M3 :0 :51 ZPJ SSE 1r 7%5 MMIMUM AME a574M-£f :3 an-iv'a a? of: ...
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